Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}2x-3y &= -3 \\ 5x-8y &= -8\end{align*}$
We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-5$ and the bottom equation by $2$ $\begin{align*}-10x+15y &= 15\\ 10x-16y &= -16\end{align*}$ Add the top and bottom equations. $-y = -1$ Divide both sides by $-1$ and reduce as necessary. $y = 1$ Substitute $1$ for $y$ in the top equation. $2x-3( 1) = -3$ $2x-3 = -3$ $2x = 0$ $x = 0$ The solution is $\enspace x = 0, \enspace y = 1$.